Surface Area 40xy²: Finding Height and Width of a Rectangular Prism

To solve this problem, we'll follow these steps:

• Step 1: Utilize the surface area formula for a rectangular prism.

• Step 2: Solve for height and width based on given surface area and length.

• Step 3: Conduct algebraic manipulations to express height and width in terms of the given variables.

Let's go through each step:

Step 1: Consider the surface area formula for a rectangular prism:

$S = 2(lw + lh + wh)$

Given the surface area $40xy^2$ and the length $l = z$, substitute these into the formula:

$40xy^2 = 2(z \cdot w + z \cdot h + wh)$

Simplifying gives us:

$20xy^2 = zw + zh + wh$

Step 2: We aim to express width $w$ and height $h$ using $x, y,$ and $z$.

By assuming one dimension as $z$, let's express the other combinations:

• Consider $w = z$. Substituting gives:

• $20xy^2 = z^2 + zh + z^2 \Rightarrow zh = 20xy^2 - 2z^2$

Thus, the height $h$ is:

$h = \frac{20xy^2 - 2z^2}{z} \Rightarrow h = 20\frac{xy^2}{z} - 2z$

Final Expression: Hence, one possible configuration for the height and width of the rectangular prism, given the surface area and length, is:

Height: $h = 10\frac{xy^2}{z}-\frac{1}{2}z$

Width: $w = z$

Therefore, the solution is option (choice 3):

$z \text{, } 10\frac{xy^2}{z}-\frac{1}{2}z$